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\section{Newton-Raphson method}
This algorithm is used to find roots of systems of non-linear equations. It is based on a simple formula $f(U + V) \approx f(U) + f'(U) + V = 0$, generalized to multivariable functions in this project. Thus, the derivate of $f$ was represented by its Jacobian Matrix.
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This algorithm should iterate until convergence to a root.But in several case, especially depending on initial vector, it diverge. To limit this phenomenon, a maximal iteration number is fixed and a backtracking system flatten the variations, preventing the norm to increase (see fig. \ref{fig1}).
%
%\begin{figure}[h]
%  \centering
%   \includegraphics[width=1\textwidth]{./../res/BacktrackingEffectH.png}
%  \caption{Backtracking Effect on g : \begin{smallmatrix}a \\ b \\ c\end{smallmatrix} $\rightarrow$\begin{smallmatrix} b*c -4 \\ a*c \\ a*b \\ a*b*c \end{smallmatrix}}
%  \label{fig1}
%\end{figure}
\begin{figure}[h]
 \begin{minipage}[b]{.5\linewidth}
 \begin{flushleft}
  \centering\epsfig{figure=./../res/BacktrackingEffectF.png,width=\linewidth}
  \caption{Algorithm convergence \label{fig1}}
  \end{flushleft}
 \end{minipage} \hfill
 \begin{minipage}[b]{.5\linewidth}
 \begin{flushright}
  \centering\epsfig{figure=./../res/BacktrackingEffectH.png,width=\linewidth}
  \caption{Backtracking improving convergence \label{fig2}}
 \end{flushright}
 \end{minipage}
\end{figure}




Figure \ref{fig1} shows that the algorithm is quickly converging to a root for $f$ : \begin{bmatrix} a \\ b\end{bmatrix}. 
\rightarrow 
\begin{bmatrix} $cos$(a) \\ $sin$(b) \end{bmatrix}
\\$and $U0 = \begin{bmatrix}1 \\ 1 \end{bmatrix}

Figure \ref{fig2} show the impact of the backtracking optimisation for $g$ : \begin{bmatrix} a \\ b \\ c\end{bmatrix}. 
\rightarrow 
\begin{bmatrix} b*c -4 \\ a*c \\ a*b \\ a*b*c \end{bmatrix}
\\$and $U0 = \begin{bmatrix}-1 \\ 1 \\ 0\end{bmatrix}


